In laser beam welding, it is usually necessary to move the welding spot at a nearly constant speed along the weld line, and to maintain the locations being welded at a constant distance from, and normal to, the laser so as to assure that the welding spot is of constant small diameter. While these relatively simple requirements are easily satisfied when the weld line is straight, they can greatly complicate the welding process when the weld line is intricately curved. A method which has been employed to weld along intricately curved weld lines, is to keep both the laser and the workpiece stationary, and to move the laser beam with mirrors. However, the apparatus required to move the welding beam is very expensive. A system which enabled welding along a weld line of a wide variety of configurations, including those which were intricately curved, using a relatively simple mechanism to achieve relative movement of the welding beam to the weld line at a substantially constant speed while maintaining the weld line at a substantially constant distance from the laser or other welding beam source, could significantly lower the cost of intricately welded parts.